Andrews–Gordon $q$-series conjecture

Let $k\geq1$, $1\leq\ell\leq k$, $1\leq i\leq k+1$, $1\leq i'\leq\ell+1$, and $|q|<1$. Let $C_k$ be the Cartan matrix of type $A_k$, let $B_{\ell-1}$ be the inverse Cartan matrix of type $A_{\ell-1}$, and let $\vec{e}_j$ be the standard basis vectors, with $\vec{e}_j=\vec{0}$ for $j\geq k+1$. Define

$$\vec{Q}_{i,i',\ell}=\vec{e}_i+\vec{e}_{i+2}+\cdots+\vec{e}_{i'}+\vec{e}_{i'+2}+\cdots+\vec{e}_{\ell+1}+\vec{e}_{\ell+3}+\cdots.$$

Let $r$ and $s$ have the meanings used in the source. Andrews–Gordon $q$-series conjecture. The parity-restricted $q$-series on the left-hand side of the statement is equal to the first displayed expression on the right when $r\equiv k\pmod2$, and to the second displayed expression when $r\not\equiv k\pmod2$. These expressions involve sums over $n$, $\vec{\mu}$, and congruence-restricted integers $j$, with the exponents and congruence conditions exactly as displayed in the source. The identities would produce new $q$-series results after replacing $q$ by $1/q$ and taking $L\to\infty$; the source gives no proof or resolution beyond presenting them as conjectures.